Question

Suppose that the time, in hours, required to repair a heat pump is a random variable X having a gamma distribution with parameters α-2 and 8 1/2. What is the probability that on the next service call (a) at most 1 hour will be required to repair the heat pump? (a) at least 2 hours will be required to repair the heat pump? (b) For the distribution in Problem 6.42 - Calculate the Expected Value, i.e. Mean (c) For the distribution in Problem 6.42 - Calculate the Variance

Answer 1

(a)
`\(P(X \leq 1) = 0.0032\) \`

(b)
`\(E(X) = 5\) hours \`

(c)
`\(Var(X) = 10\) hours²`

Step-by-step explanation:

(a) To find the probability that at most 1 hour is required to repair the heat pump with a gamma distribution, we can use the cumulative distribution function
`(CDF)`. For a gamma distribution with parameters
`\(\alpha-2\)`and
`\(8 \, 1/2\)`, the probability
`\(P(X \leq 1)\)`is approximately 0.0032.

(b) The expected value or mean
`(\(E(X)\))`for a gamma distribution with parameters
`\(\alpha\)` and
`\(\beta\)`is given by
`\(E(X) = \alpha/\beta\)`. In this case, with parameters
`\(\alpha-2\)` and
`\(8 \, 1/2\)`, the expected time to repair the heat pump is 5 hours.

(c) The variance
`(\(Var(X)\))`for a gamma distribution with parameters
`\(\alpha\)` and
`\(\beta\)` is given by
`\(Var(X) = \alpha/\beta^2\)`. Using the given parameters
`\(\alpha-2\)`and
`\(8 \, 1/2\)`, the variance for the time to repair the heat pump is 10 hours². The variance provides a measure of the spread or variability in the distribution.